Let us begin a systematic study: if we put \(\delta _1=\delta _2\)  in Eq.(REF ) and solve it for \(\kappa _2\) , we find three real solutions, \(\kappa _2=\kappa _1\)  and \(\kappa _2=\pm \sqrt{4-\kappa _1^2}\) . We plot these solutions in Fig.(REF ). In practice, on the entire parameter space of \(\kappa _1\)  and \(\kappa _2\) , these solutions give us the specific locus where the growth-rates of the breathers are equal. The first solution, \(\kappa _1=\kappa _2\)  is the so-called degenerate solution condition where the solution becomes undefined. However, using l'Hôspital's rule, the solution can be recovered even in this case [1]}, [2]}.
